Chapter 3.6, Problem 3E

  $\left(a\right)$

To determine

To calculate: The value of trigonometric expression $\arctan \left(1\right)$ .

Answer to Problem 3E

The value of trigonometric expression $\arctan \left(1\right)$ is $\frac{\pi }{4}$ .

Explanation of Solution

Given information:

The trigonometric expression $\arctan \left(1\right)$ .

Formula used:

Inverse tangent trigonometric function is expressed as ${\tan }^{-1}\left(x\right)=y$ which is equivalent to $x=\tan y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Calculation:

Consider trigonometric expression $\arctan \left(1\right)$ .

Recall that the tangent trigonometric function is expressed as ${\tan }^{-1}\left(x\right)=y$ which is equivalent to $x=\tan y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Recall that value of tangent function at an angle $\frac{\pi }{4}$ is $1$ .

Also the angle $\frac{\pi }{4}$ lies between $-\frac{\pi }{2}\text{ and }\frac{\pi }{2}$ .

That is, $\tan \left(\frac{\pi }{4}\right)=1$ .

Take sine function to the right hand side,

  ${\tan }^{-1}\left(1\right)=\frac{\pi }{4}$ .

Thus, the value of trigonometric expression $\arctan \left(1\right)$ is $\frac{\pi }{4}$ .

  $\left(b\right)$

To determine

To calculate: The value of trigonometric expression ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ .

Answer to Problem 3E

The value of trigonometric expression ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ is $\frac{\pi }{4}$ .

Explanation of Solution

Given information:

The trigonometric expression ${\sec }^{-1}\left(2\right)$ .

Formula used:

Inverse sine trigonometric function is expressed as ${\sin }^{-1}\left(x\right)=y$ which is equivalent to $x=\sin y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Calculation:

Consider trigonometric expression ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ .

Recall that the inverse sine trigonometric function is expressed as ${\sin }^{-1}\left(x\right)=y$ which is equivalent to $x=\sin y$ where $-\frac{\pi }{2}<y<\frac{\pi }{2}$ .

Recall that value of sine function at an angle $\frac{\pi }{4}$ is $\frac{1}{\sqrt{2}}$ .

Also the angle $\frac{\pi }{4}$ lies between $-\frac{\pi }{2}\text{ and }\frac{\pi }{2}$ .

That is, $\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$ .

Take cosine function to the right hand side,

  ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi }{4}$ .

Thus, the value of trigonometric expression ${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$ is $\frac{\pi }{4}$ .